By Professor Luigi Accardi, Professor Igor Volovich, Professor Yun Gang Lu (auth.)
ISBN-10: 3642075436
ISBN-13: 9783642075438
ISBN-10: 3662049295
ISBN-13: 9783662049297
The topic of this ebook is a brand new mathematical process, the stochastic restrict built for fixing nonlinear difficulties in quantum idea concerning platforms with infinitely many levels of freedom (typically quantum fields or gases within the thermodynamic limit). this method is condensed into a few simply utilized principles (called "stochastic golden rules") which permit to unmarried out the dominating contributions to the dynamical evolution of platforms in regimes related to lengthy instances and small results. within the stochastic restrict the unique Hamiltonian concept is approximated utilizing a brand new Hamiltonian conception that is singular. those singular Hamiltonians nonetheless outline a unitary evolution and the recent equations supply even more perception into the suitable actual phenomena than the unique Hamiltonian equations. specially, one could explicitly compute multi-time correlations (e.g. photon information) or coherent vectors, that are past the achieve of common asymptotic recommendations in addition to deduce within the Hamiltonian framework the generally used stochastic Schrödinger equation and the grasp equation. This monograph is definitely acceptable as a textbook within the rising box of stochastic restrict innovations in quantum theory.
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Additional resources for Quantum Theory and Its Stochastic Limit
Example text
3) converges to a Brownian motion by the effect of a functional central limit theorem. 26 Aigebraic Formulation of the Stochastic Limit The present section is not necessary for the understanding of the rest of the book. In it we formulate in an abstract algebraic setting the stochastic limit problem. 26 Algebraic Formulation of the Stochastic Limit 47 Let A~ be a C*-algebra (algebra of observables). We assume that A>. ) where 1l>. is a Hilbert space (always complex separable) and A E R is a parameter (in some cases A E R 2 , R 3 , ••• ).
These coefficients shall play an important role in the following. 2. 6) is essentially the only one which guarantees the existence and nontriviality of the limit of the vacuum expectation of the second-order of the series term. The limit (1. 6) means that in times of order t / >.. 2 the interaction produces efIects of order T. Thus >.. provides a natural time scale for the observable efIects of the interaction. 2t -+ T {:} t rv T / >.. 6); and (ii) only for the second-order term. (l) (ii) If the answer to (i) is affirmative, can we conclude that the term-byterm limit of the time-rescaled perturbative series can be re-summed into a new evolution operator, Ut ?
The explicit form of L * is easy calculated using the identity and this leads to the master equation: Thus, given PR and the interaction Hamiltonian, the explicit form of the master equation can be calculated. More on the master equation can be found in Seets. 14. 9) and using the same change of variables, we find ([H1(0), [H1(t2),b(,\2 t2 +tl)]DR = (H1(0)H1(t2)b(,\2 t2 + tt))R - (H1(t2)b(,\2 t2 + tdH1(0))R - (HI(0)b(,\2 t2 + tl)HI(h))R + (b(,\2 t2 + tdH1(t2)H1(0))R. 2t 2 + h))R (Hr(0)b(),2t2 + tl)Hr(t2))R = (Hr(O) (b(>h 2 + h))RHr(t2))R (Hr(0)Hr(t2)b(),2t2 (and the same relations for the adjoint).
Quantum Theory and Its Stochastic Limit by Professor Luigi Accardi, Professor Igor Volovich, Professor Yun Gang Lu (auth.)
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